Phase-space integration moment-transport equation

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

In summary, computing the moment-transport equations starting from Eq. (4.39) involves integration over phase space using the mles described above for particular choices of g. In the following, we will assume that the flux term at the boundary of phase space can be neglected. However, the reader should keep in mind that this assumption must be verified for particular cases. 

In Section 4.3, example macroscale transport equations are derived for selected moments of the NDF. Having introduced the precise forms of the mesoscale advection models in Eq. (5.2), it is of interest to derive explicitly some example moment source terms resulting from these models. In order to do so, we will use the rules presented in Section 4.3.1 for phase-space integration. For simplicity, we consider only the advection term involving (Afp)i and assume that the only phase-space variables of interest are v and Vf, and that the model in Eq. (5.2) reduces to... 

The extension of PTC to higher spatial dimensions is straightforward and many examples can be found in the literature (see, for example, Desjardins et al. (2008)). In Figure 8.1 we show a 2D example with two jets crossing. An important question that arises with PTC is whether or not the moment-transport equations can predict it. To answer this question, we return to Eq. (8.3) and integrate over phase space to find the moment-transport equation ... 

Let us now discuss in detail the question of moment conservation during time integration. Consistently with Chapter 8, the source terms due to phase-space processes are set to zero so that only transport terms in real space are considered in this discussion. When Eq. (D.23) is integrated using an explicit Euler scheme, the volume-average moment of order k in the cell centered at X at time (n + l)Af is directly calculated from the volume-average moment of order k at time n Af from the following equation ... 

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